Logarithms
Let’s Get It Started!!!
Remember
A
logarithm is an exponent
 Every
time you are working
with logarithms, you can
substitute the word exponent
for logarithm,
The symbol for logarithm is
log
What is written
log 3 9
What is said
log to the base 3 of 9
or
log base 3 of 9
What does it mean????
log 3 9
means
what power of 3 is 9?
log 3 9 = 2 because 3 2 = 9
^
^
logarithmic form
exponential form
Write in exponential form:
Logarithmic Form
log 2 8 = 3
Exponential Form
23=8
log 6 36 = 2
6 2 = 36
log 4 1 = 0
4 0= 1
log 2 32 = 5
2 5 = 32
Write in logarithmic form
Exponential Form
Logarithmic Form
3
0
= 1
log
3
1= 0
5
4
= 625
log
5
625 = 4
= .1
log
10
log
6
10
-1
6
= 216
3
(.1) = -1
216 = 3
Special log values
log
a
1=0
because
a
0
=1
log
a
a=1
because
a
1
=a
log
a
a
a
x
=a
x
= x because
x
We interrupt the lesson on logarithms to
bring you an important message…
Hold on!!
Let’s make sure you
understand what “if and
only if” means.
A triangle is scalene if and only if the
triangle has 3 sides of different lengths.
The above statement takes the place of 2 statements:
A triangle is scalene if it has 3 sides of different
lengths.
AND
A triangle with 3 sides of different lengths is scalene.
Now back to the lesson on
logarithms…
Definition
The logarithm of x with base a (a > 0) is denoted by
log a x and is defined as
y = log a x iff
ay=x
if and only if
y = log a x
Replace the y with f (x) and you have
a function
f ( x ) = log
a
x
Evaluate:
log
5
25 =
2
log
4
64 =
3
log
2
64 =
6
log
4
2=
½
log
3
(-1) =
Impossible Why????
The log function with base 10 is called the
common log function
Your calculator uses base 10 when doing
logarithms.
If using common logs, you do not need to
write in the base of 10.
log 100 = 2
because
10
2
= 100
If you have any other base, to use your
calculator, you can use the CHANGE OF
BASE FORMULA:
log a x =
log x / log a
Use your calculator to evaluate:

log 4 7 =
log 7 / log 4 
1.404
Log 3 40 =
log 40 / log 3 
3.358
More Examples using the calculator
log
3
543 =
log 543 / log 3  5.732
log
4
84 =
log 84 / log 4  3.196
log
2
79 =
log 79 / log 2  6.304
Graph f ( x ) = log 2 x
x
Make a table of values
y
1
0
2
1
3
1.6
4
2
8
3
.5
-1
.25
-2
Let’s review logarithms
A logarithm is an exponent
Calculators use base 10
To find log 5 399, using
a calculator, use
log 399 / log 5 (  3.72)
The End!
HOMEWORK
TIME!!!